From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: approximation for g_2(L),g_3(L) (L=Lattice)
Date: Thu, 26 Aug 1999 12:33:44 GMT
Newsgroups: sci.math

In article <37C514E4.AC14B094@stud.uni-hannover.de>,
  Stephan Bauer <Stephan.Bauer@stud.uni-hannover.de> wrote:
> Hi.
>
> I want to calculate g_2(L) and g_3(L) where L (subset of C) is a
> lattice  and
> 	g_2= 60 * sum_{w in L\{0}} 1/w^4
> 	g_3= 140 * sum_{w in L\{0}} 1/w^6.
> I am looking for a numerical approximation  for g_2 and g_3
> (references).

These are standard examples of modular forms. Let's assume that
L has generators 1 and w where Im(w) > 0. [The general case
comes easily from this.] In Serre's "A course in arithmetic"
one finds the formulas
g_2 = (4 pi^4/3) E_2
and
g_3 = (8 pi^6/27) E_3
where E_2 = 1 + 240 sum_{n=1}^infinity sigma_3(n) q^n and
E_3 = 1 - 504 sum_{n=1}^infinity sigma_5(n) q^n.
Here q = exp(2 pi i w) and sigma_j(n) is the sum of the j-th powers
of the positive integer divisors of n. One can ensure that
Im(w) >= 1/2 and so the convergence will be reasonably fast.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "They did not have proper palms at home in Exeter."
    Peter Carey, _Oscar and Lucinda_


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